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(ELEC214)2002fallproblem1.pdf
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ELEC 214
Communications Systems
Test 1 C October 10, 2002

Time allowed: 1 hour 20 minutes Students should answer ALL 3 Questions Total 100 points Note: This is a closed book test.
Question 1 (33 points)
1. Let x(t) = a x1(t) + b x2(t), where x1(t) = A1 cos(1t + 1) and x2(t) = A2 cos(2t + 2).
(a)
Find the condition(s) under which x(t) is periodic. (5 points)

(b)
Let x(t) = 2cos(280t) + 3sin(700t) . Is x(t) periodic? Please explain your answer. (5 points)


2. Find the energy spectral densities of the following signals.
(a)
x1(t) = 20sinc(30t) (4 points)

(b)
Find the total energy in x1(t) (3 points)

(c)
x2(t) = 4(5t)cos(40t) . (4 points)


3. For x(t) = A cos(2 f0 t + ),
(a)
Find the time autocorrelation function of x(t). (4 points)

(b)
Find the power spectral density function of x(t). (4 points)

(c)
Find the total power of x(t). (4 points)


Question 2 (33 points)
4. An AM modulator has output
xc (t) = 20cos(2150t) + 6cos(2160t) + 6cos(2140t)

(a)
Determine the modulation index. (4 points)

(b)
Determine the efficiency. (4 points)

(c)
Give two possible methods for demodulation. (5 points)


(d) What are the pros and cons of the two methods? (5 points)
5. For m(t) =6 cos(40 t) + 10 cos(100 t) and an AM modulator with c(t) = 20 cos(1000 t) and a modulation index of 0.5.
(a)
Give the expression of the modulated carrier, xc(t). (5 points)

(b)
Find the efficiency of the modulator. (5 points)

(c)
Sketch the double-sided magnitude spectrum of xc(t), the modulator output. Label the amplitude and frequency of each spectral component.



(5 points)
Question 3 [34 points]
A receiver uses an IF frequency of 400 kHz as shown in Figure 1. The receiver is tuned to a transmitter having a carrier frequency of 1200 kHz. Therefore, the permissible frequencies of the local oscillator are (1200 400) Hz.


Figure 1 Block diagram of the receiver
(a)
Figure 2 shows the spectrum allocations for two AM radio broadcasting stations. Radio Station 1 is allocated the spectrum from 1100 to 1300 kHz while the spectrum allocated to the Radio Station 2 is from 1900 to 2100 kHz. Assume that the all the phase components are zeros.


Figure 2 Spectrum of two radio stations in double-sided representation.
The above receiver is used to translate the broadcasted signal in Radio Station 1 to IF band. In this part, the higher permissible frequency is selected to be the frequency of the local oscillator.
(i) Sketch the spectrum at IF filter input. Please label all the frequencies and shaded spectrum clearly. (9 points)
(ii) What are the appropriate values of the bandwidth and center frequency of the IF filter? (4 points)
(iii) Sketch the spectrum at IF filter output. (3 points)

(b) Figure 3 shows another spectrum allocations for two AM radio broadcasting stations. The Radio Station 1 allo